What is the integral of 1/2 sin(2x)e^{sin(x)} ?

The integral of 1/2 sin(2x)e^{sin(x)} is e^{sin(x)}sin(x)-e^{sin(x)}+C

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integral of 1/2 sin(2x)e^{sin(x)}

\int \frac{1}{2} sin (2 x) e^{s i n (x)} d x

e^{s i n (x)} sin (x) - e^{s i n (x)} + C

Solution steps

\int \frac{1}{2} sin (2 x) e^{s i n (x)} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin (2 x) e^{s i n (x)} d x

Rewrite using trig identities

= \frac{1}{2} \cdot \int e^{s i n (x)} \cdot 2 cos (x) sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int e^{s i n (x)} cos (x) sin (x) d x

Apply Integration By Parts

: sin (x) e^{s i n (x)} - \int cos (x) e^{s i n (x)} d x

= \frac{1}{2} \cdot 2 (sin (x) e^{s i n (x)} - \int cos (x) e^{s i n (x)} d x)

\int cos (x) e^{s i n (x)} d x = e^{s i n (x)}

= \frac{1}{2} \cdot 2 (sin (x) e^{s i n (x)} - e^{s i n (x)})

Simplify

\frac{1}{2} \cdot 2 (sin (x) e^{s i n (x)} - e^{s i n (x)}) : e^{s i n (x)} sin (x) - e^{s i n (x)}

= e^{s i n (x)} sin (x) - e^{s i n (x)}

Add a constant to the solution

= e^{s i n (x)} sin (x) - e^{s i n (x)} + C

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What is the integral of 1/2 sin(2x)e^{sin(x)} ?
The integral of 1/2 sin(2x)e^{sin(x)} is e^{sin(x)}sin(x)-e^{sin(x)}+C